5. The decay constant for a particular radioactive element is 0.015 when time is measured in years. Find the half-life of this element.
5. The decay constant for a particular radioactive element is 0.015 when time is measured in years. Find the half-life of this element.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem 5: Calculating the Half-Life of a Radioactive Element
**Problem Statement:**
The decay constant for a particular radioactive element is 0.015 when time is measured in years. Find the half-life of this element.
**Solution Explanation:**
To find the half-life (\( t_{1/2} \)) of a radioactive element, we can use the formula that relates the decay constant (\( \lambda \)) to the half-life:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
Where:
- \( \ln(2) \) is the natural logarithm of 2.
- \( \lambda \) is the decay constant.
In this case, the given decay constant (\( \lambda \)) is 0.015 per year.
### Steps:
1. **Calculate the natural logarithm of 2:**
\[ \ln(2) \approx 0.693 \]
2. **Divide the natural logarithm of 2 by the decay constant \( \lambda \):**
\[ t_{1/2} = \frac{0.693}{0.015} \]
3. **Perform the division:**
\[ t_{1/2} \approx 46.2 \]
### Result:
The half-life of the radioactive element is approximately 46.2 years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3c4489a-e4bd-4ff1-89b5-d5833f06911e%2F24a20916-14fb-4d58-ac20-ee7208a3b593%2Fnhd0u4_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 5: Calculating the Half-Life of a Radioactive Element
**Problem Statement:**
The decay constant for a particular radioactive element is 0.015 when time is measured in years. Find the half-life of this element.
**Solution Explanation:**
To find the half-life (\( t_{1/2} \)) of a radioactive element, we can use the formula that relates the decay constant (\( \lambda \)) to the half-life:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
Where:
- \( \ln(2) \) is the natural logarithm of 2.
- \( \lambda \) is the decay constant.
In this case, the given decay constant (\( \lambda \)) is 0.015 per year.
### Steps:
1. **Calculate the natural logarithm of 2:**
\[ \ln(2) \approx 0.693 \]
2. **Divide the natural logarithm of 2 by the decay constant \( \lambda \):**
\[ t_{1/2} = \frac{0.693}{0.015} \]
3. **Perform the division:**
\[ t_{1/2} \approx 46.2 \]
### Result:
The half-life of the radioactive element is approximately 46.2 years.
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